Question

Prove that
$\sqrt[3]{11}$
is irrational


​

Answer 1

Step-by-step explanation:

Lets assume that $\sqrt[3]{11}$ is rational

A rational number is represented by $\frac{p}{q}$ (p and q are positive integers and are relatively prime (also known as coprime)

$\sqrt[3]{11}$  =   $\frac{p}{q}$

Taking cubes on both sides

11 = $\frac{p^3}{q^3}$

11q³ = p³

Let p = 11N

now, 11q³ = 11³N³

now, we see that p is divisible by 11, thus, p is a factor of 11

but this contradicts the fact that p and q are co- prime

so, this number can't be rational

∴ $\sqrt[3]{11}$ is irrational

Answer 2

Answer:

Lets assume that \sqrt[3]{11}

3

11

is rational

A rational number is represented by \frac{p}{q}

q

p

(p and q are positive integers and are relatively prime (also known as coprime)

\sqrt[3]{11}

3

11

= \frac{p}{q}

q

p

Taking cubes on both sides

11 = \frac{p^3}{q^3}

q

3

p

3

11q³ = p³

Let p = 11N

now, 11q³ = 11³N³

now, we see that p is divisible by 11, thus, p is a factor of 11

but this contradicts the fact that p and q are co- prime

so, this number can't be rational

∴ \sqrt[3]{11}

3

11

is irrational